No offense to prisms and cylinders. They're great. But we don't eat our ice cream out of prisms while touring the Egyptian Cylinders of Giza. What we're saying is that there's no substitute for pyramids and cones, so we might as well learn a little bit about them.
A pyramid is a regular pyramid if the base is a regular polygon with a vertex that's perpendicular to the center of the base. A pyramid is irregular if it doesn't have daily bowel movements.
The height of the overall pyramid is called the altitude, but the lateral faces have their own height called the slant height. Let's take a regular hexagonal pyramid and look at its net (which looks surprisingly like the Star of David. L'chaim!).
The lateral area is just the area of all the triangles combined. The area of a triangle is half the base times the height. In this case, the base is the side of the hexagon s and the height of the triangle is the slant height l. Since we have six of these triangles, the lateral area is given by this equation:
Or, simplified, it equals:
Since we know the perimeter of the hexagon equals 6s, we can replace the sides with the perimeter of the hexagon P.
Mazel tov! We've found the lateral area formula for any pyramid.
All that's missing for us to find the surface area is the base of the pyramid, B. Remember that unlike cylinders and prisms, pyramids only have one base. If we know the area of the base, we can find out the surface area of the whole pyramid.
Sample Problem
For Father's Day, you decide to build your dad a birdhouse. You know how much he loves birds, and you'd rather do that than be his golf caddy for the millionth time. (Seriously, how many times can he call you "champ" in one day?) You decide that roof of the birdhouse will be a square pyramid because it's the easiest. The slant height of the roof is 10 inches and each side of the square is 8 inches. How much wood in square inches will you need to make the roof?
Well, we can start with our lateral area formula.
The perimeter of the square base is 8 + 8 + 8 + 8 = 32 inches. The slant height that you want for the roof is 10 inches.
Just be careful with that hacksaw. You don't want to end up like that woodshop teacher with the missing pinky.
We can use far more than these formulas to figure out surface area. Remember the Pythagorean Theorem? Remember trigonometry? Those mathematical ghosts will never leave you. As long as we treat them right, they'll be more like Casper and less like the Exorcist.
Sample Problem
What is the surface area of this pentagonal pyramid?
Knowing the altitude and the slant height can give us enough information to calculate everything we need. If we look at those two values as a right triangle, the altitude is a leg and the slant height is the hypotenuse. Solving for the remaining leg (and hopefully it isn't prosthetic) will give us the distance from the center of the base to one of the sides.
a2 + b2 = c2
a2 + 122 = 132
a2 = 169 – 144
a2 = 25
a = 5
Quick tangent: the distance from the center of any regular polygon to the midpoint of one of its edges is called the apothem, which is what we just found.
Okay, tangent over. Now zoom in on the base of our pyramid. A perpendicular line segment from the center to the side is 5 inches (that's the apothem). Since the pentagon is regular, we can also find the angles in the right triangle. The 360° around the center is split into 5 evenly, which gives us 72° per isosceles triangle. We're splitting these triangles in half, so each angle is 36°. Here's what we know, Pictionary style.
Hey, speaking of going off on a tangent...
We can solve for s and find the length of each side of the pentagon.
s = 10tan(36°) ≈ 7.27 inches
Be sure to thank trigonometry in your Nobel Prize acceptance speech.
Now we can do two things:
- Find the area of the pentagonal base
- Find the perimeter, which will help us find the lateral area
Area is first, so we'll start with that.
If you imagine cutting the pentagon into 5 identical triangles (feel free to imagine cutting it a little more if you're really mad at it), we can find the area of each triangle and multiply it by 5. The base of the triangle is the side length and the height is 5.
Halfway done. If this were a romantic comedy, we'd be at the point where the two main characters have fallen in love already but don't know it yet, and some random conflict has just happened that tears them apart. That leaves just enough time for the epiphany of their love, the frantic last-minute chase for each other, and the obligatory saliva swap at the end.
The base is a pentagon ("penta-" means five), so perimeter is five times the length of the side. We know the side is about 7.27 inches, which means the perimeter is about 36.35 inches. High penta!
Now for the part where the two main characters realize their undying love and race to find each other. The lateral area of a pyramid is given by the formula:
We know the slant height l is 13 since it was given to us, and we just calculated P.
L = ½(36.35 × 13) ≈ 236.2 in2
The two lovers have reunited and professed their love at long last. All that remains is that slobbery kiss that lasts for centuries.
SA = L + B
SA = 236.2 in2 + 90.8 in2 = 327 in2
The end. Roll credits to some feel-good Michael Bolton song.